combining the above , we might be able to compute f(x) such that
f(f(x)) = exp(x)

or slog(x).

and maybe also arrive at the last uniqueness conditions , in the sence of a single unique slog(z) as the general solution for tetration

slog(z) + C

( C some constant )

thats the main idea , i dont expect the function to be entire however analytic continuation might be possible , although that may loose the conditions above on the domain where the continuation takes place ( but who cares ).

if analytic continuation is not possible than probably a real analytic one is still possible ... i think ...